"... Abstract. For an operator, A, with cyclic vector ϕ, we study A + λP where P is the rank one projection onto multiples of ϕ. If [α, β] ⊂ spec(A) andA has no a.c. spectrum, we prove that A + λP has purely singular continuous spectrum on (α, β) for a dense Gδ of λ’s. The subject of rank one perturbati ..."

Abstract. For an operator, A, with cyclic vector ϕ, we study A + λP where P is the rank one projection onto multiples of ϕ. If [α, β] ⊂ spec(A) andA has no a.c. spectrum, we prove that A + λP has purely singular continuous spectrum on (α, β) for a dense Gδ of λ’s. The subject of rank one perturbations of self-adjoint operators and the closely related issue of the boundary condition dependence of Sturm-Liouville operators on [0, ∞) has a long history. We’re interested here in the connection with Borel-Stieltjes transforms of measures (Im z&gt;0):

...ND LOCALIZATION 25 bound on 〈x 2n 〉(t), regardless of whether H has SULE, orevenwhetherH has only pure pointspectrumornot. 2. By a result of Last [22], which extends an idea originaly due to Guarneri =-=[12]-=-, it follows that if the spectral measure of δ0 (for Hλ) is not supported on a set of Hausdorff dimension zero, then for some β>0, lim t −2nβ 〈x 2n 〉(t) > 0. Thus, we get an alternative proof to the f...

. We study relations between quantum dynamics and spectral properties, concentrating on spectral decompositions which arise from decomposing measures with respect to dimensional Hausdorff measures. 1. Introduction Let H be a separable Hilbert space, H : H ! H a self adjoint operator, and / 2 H (with k/k = 1). The spectral measure ¯/ of / (and H ) is uniquely defined by [24]: h/ ; f(H)/i = Z oe(H) f(x) d¯/ (x) ; (1:1) for any measurable (Borel) function f . The time evolution of the state / , in the Schrodinger picture of quantum mechanics, is given by /(t) = e \GammaiHt / : (1:2) The relations between various properties of the spectral measure ¯/ (with an emphasis on &quot;fractal&quot; properties) and the nature of the time evolution have been the subject of several recent papers [7,13,15--18,20,22,33,36,39]. Our purpose in this paper is twofold: First, we use a theory, due to Rogers and Taylor [28,29], of decomposing singular continuous measures with respect to Hausdorff measures to i...

...quantum dynamics to spectral properties. Moreover, the decomposition theory allows us to extend some of these results, and, in particular, we get a strengthened version of the Guarneri-Combes theorem =-=[7,15]-=-. While most of our discussion of quantum dynamics applies to any self-adjoint Hamiltonian, and is therefore rather general; the primary example we have in mind is that of a single electron in an exte...

"... We consider discrete one-dimensional Schrodinger operators with quasi-Sturmian potentials. We present a new approach to the trace map dynamical system which is independent of the initial conditions and establish a characterization of the spectrum in terms of bounded trace map orbits. Using this, ..."

We consider discrete one-dimensional Schrodinger operators with quasi-Sturmian potentials. We present a new approach to the trace map dynamical system which is independent of the initial conditions and establish a characterization of the spectrum in terms of bounded trace map orbits. Using this, it is shown that the operators have purely singular continuous spectrum and their spectrum is a Cantor set of Lebesgue measure zero. We also exhibit a subclass having purely ff-continuous spectrum. All these results hold uniformly on the hull generated by a given potential.

...volution of the associated quantum systems can be obtained by means of certain Hausdorff dimensional properties of the spectral measures. The first results in this direction were obtained by Guarneri =-=[14]-=- and Combes [5]. They essentially required uniform α-Hölder continuity. Their results were extended by Last in [32] to spectral measures with non-trivial α-continuous component, that is, measures that...

"... A random polymer model is a one-dimensional Jacobi matrix randomly composed of two finite building blocks. If the two associated transfer matrices commute, the corresponding energy is called critical. Such critical energies appear in physical models, an example being the widely studied random dimer ..."

A random polymer model is a one-dimensional Jacobi matrix randomly composed of two finite building blocks. If the two associated transfer matrices commute, the corresponding energy is called critical. Such critical energies appear in physical models, an example being the widely studied random dimer model. It is proven that the Lyapunov exponent vanishes quadratically at a generic critical energy and that the density of states is positive there. Large deviation estimates around these asymptotics allow to prove optimal lower bounds on quantum transport, showing that it is almost surely over-diffusive even though the models are known to have pure-point spectrum with exponentially localized eigenstates for almost every configuration of the polymers. Furthermore, the level spacing is shown to be regular at the critical energy.

... 4) given the boundedness of transfer matrices (Theorem 7) was suggested by S. Tcheremchantsev. This technique, which will be published in full generality in [DT], is simpler than the Guarneri method =-=[Gua]-=- of proving lower bounds employed in [JSS] (and also applicable here) and allowed us to circumvent previous more intricate arguments. We greatly appreciate that S. Tcheremchantsev made his work availa...

"... We derive a general upper bound on the spreading rate of wavepackets in the framework of Schrödinger time evolution. Our result consists of showing that a portion of the wavepacket cannot escape outside a ball whose size grows dynamically in time, where the rate of this growth is determined by prope ..."

We derive a general upper bound on the spreading rate of wavepackets in the framework of Schrödinger time evolution. Our result consists of showing that a portion of the wavepacket cannot escape outside a ball whose size grows dynamically in time, where the rate of this growth is determined by properties of the spectral measure and by spatial properties of solutions of an associated time independent Schrödinger equation. We also derive a new lower bound on the spreading rate, which is strongly connected with our upper bound. We apply these new bounds to the Fibonacci Hamiltonian - the most studied one-dimensional model of quasicrystals. As a result, we obtain for this model upper and lower dynamical bounds establishing wavepacket spreading rates which are intermediate between ballistic transport and localization. The bounds have the same qualitative behavior in the limit of large coupling.

.... This question, “What determines the spreading of a wavepacket” [24], has been an active field of research over the last 15 years and there is by now a considerable body of literature devoted to it (=-=[1, 2, 3, 4, 9, 10, 14, 15, 16, 17, 18, 19, 20, 24, 25, 27, 28, 29, 30, 33, 34, 41, 42, 44]-=- is just a partial list). We note that there are some situations, such as some systems with absolutely continuous spectrum which can be studied by scattering theory [32], where good understanding of t...

"... We consider ergodic families of Schrödinger operators over base dynamics given by strictly ergodic subshifts on finite alphabets. It is expected that the majority of these operators have purely singular continuous spectrum supported on a Cantor set of zero Lebesgue measure. These properties have in ..."

We consider ergodic families of Schrödinger operators over base dynamics given by strictly ergodic subshifts on finite alphabets. It is expected that the majority of these operators have purely singular continuous spectrum supported on a Cantor set of zero Lebesgue measure. These properties have indeed been established for large classes of operators of this type over the course of the last twenty years. We review the mechanisms leading to these results and briefly discuss analogues for CMV matrices.

...m and the RAGE theorem. Last also addressed this issue in [105] and proposed a decomposition of spectral measures with respect to Hausdorff measures. This was motivated by earlier results of Guarneri =-=[72]-=- and Combes [28] who proved dynamical lower bounds for initial states with uniformly Hölder continuous spectral measures. By approximation with uniformly Hölder continuous measures, Last proved in [10...

"... We present an approach to quantum dynamical lower bounds for discrete one-dimensional Schrodinger operators which is based on power-law bounds on transfer matrices. It suces to have such bounds for a nonempty set of energies. We apply this result to various models, including the Fibonacci Hamil ..."

We present an approach to quantum dynamical lower bounds for discrete one-dimensional Schrodinger operators which is based on power-law bounds on transfer matrices. It suces to have such bounds for a nonempty set of energies. We apply this result to various models, including the Fibonacci Hamiltonian.

...ime-averaged moments which take into account (in a somewhat hidden form) polynomial tails, are obtained in [3] and [37]. The proofs are based on the spectral theorem and develop the ideas of Guarneri =-=[19]-=-. The obtained lower bounds are expressed in terms of spectral measure µψ [3, 37] or in terms of both spectral measure and generalized eigenfunctions uψ(n, E) [37]. To apply them to concrete quantum s...

"... 1. Introduction and main results. In this paper we investigate the relations between the rate of decay of solutions of Schrödinger equations, continuity properties of spectral measures of the corresponding operators, and dynamical properties of the corresponding quantum systems. The first main resul ..."

1. Introduction and main results. In this paper we investigate the relations between the rate of decay of solutions of Schrödinger equations, continuity properties of spectral measures of the corresponding operators, and dynamical properties of the corresponding quantum systems. The first main result of this paper shows that, in great generality, certain upper bounds on the rate of growth of L2 norms of generalized

...nuous spectral subspace, the set of all vectors ξ such that µ ξ is α-continuous (see [30]). In particular, if µ ψ has an α-continuous component (i.e., Pαcψ �= 0), then the following lower bound holds =-=[8, 16, 17, 30]-=-: 〈〈|X| m 〉〉T ≥ CmT mα d (here d is the space dimension and Cm is a constant depending on µ ψ and m). Recall that for a wide class of Schrödinger operators, one has a generalized eigenfunction expansi...

"... Abstract. We develop a general method to bound the spreading of an entire wavepacket under Schrödinger dynamics from above. This method derives upper bounds on time-averaged moments of the position operator from lower bounds on norms of transfer matrices at complex energies. This general result is a ..."

Abstract. We develop a general method to bound the spreading of an entire wavepacket under Schrödinger dynamics from above. This method derives upper bounds on time-averaged moments of the position operator from lower bounds on norms of transfer matrices at complex energies. This general result is applied to the Fibonacci operator. We find that at sufficiently large coupling, all transport exponents take values strictly between zero and one. This is the first rigorous result on anomalous transport. For quasi-periodic potentials associated with trigonometric polynomials, we prove that all lower transport exponents and, under a weak assumption on the frequency, all upper transport exponents vanish for all phases if the Lyapunov exponent is uniformly bounded away from zero. By a well-known result of Herman, this assumption always holds at sufficiently large coupling. For the particular case of the almost Mathieu operator, our result applies for coupling greater than two. 1.

...sure, defined as the unique Borel measure obeying ∫ 〈ψ, f(H)ψ〉 = f(E) dµψ(E) σ(H) for every measurable function f. Here, 〈·, ·〉 denotes the scalar product in H. A very important discovery of Guarneri =-=[39, 40]-=-, which has been extended by other authors [14, 41, 62, 58], was that suitable continuity properties of the spectral measure dµψ imply lower bounds on the spreading of the wavepacket. Such continuity ...